A subset $S$ of a metric space $(M, d)$ is bounded if it is contained in a ball of finite radius, i.e. if there exists $x$ in $M$ and $r > 0$ such that for all $s$ in $S$, we have $d(x, s) < r$.

If I understand it correctly, the first definition requires that the origin of the "open disc" must be inside the subset, whereas the second definition does not have this restriction. Are these definitions somehow the same, or are they different? If not which one is correct?

The motivation for this question is because I want to understand what the Heine-Borel Theorem means.

4 Answers
4

They are the same. Say you have any $x$ with $M \subset B_r(x)$, where $B_r(x) = \{y \,:\, d(x,y) < r\}$. Then pick an arbitrary $z \in M$. Due to the triangle inequality, you have $$
M \subset B_{r + d(x,z)}(z) \text{.}
$$